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A057204
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Primes congruent to 1 mod 6 generated recursively. Initial prime is 7. The next term is p(n) = Min_{p is prime; p divides 4Q^2+3; p mod 6 = 1}, where Q is the product of previous entries of the sequence.
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28
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7, 199, 7761799, 487, 67, 103, 3562539697, 7251847, 13, 127, 5115369871402405003, 31, 697830431171707, 151, 3061, 229, 193, 5393552285540920774057256555028583857599359699, 709, 397, 37, 61, 46168741, 3127279, 181, 122268541
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OFFSET
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1,1
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COMMENTS
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4*Q^2 + 3 always has a prime divisor congruent to 1 modulo 6.
If we start with the empty product Q=1 then it is not necessary to specify the initial prime. - Jens Kruse Andersen, Jun 30 2014
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REFERENCES
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P. G. L. Dirichlet (1871): Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, Supplement VI, 24 pages.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 13.
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LINKS
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EXAMPLE
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a(4)=487 is the smallest prime divisor of 4*Q*Q + 3 = 10812186007, congruent to 1 (mod 6), where Q = 7*199*7761799.
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MATHEMATICA
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a={7}; q=1;
For[n=2, n<=7, n++,
q=q*Last[a];
AppendTo[a, Min[Select[FactorInteger[4*q^2+3][[All, 1]], Mod[#, 6]==1 &]]];
];
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PROG
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(PARI) Q=1; for(n=1, 11, f=factor(4*Q^2+3); for(i=1, #f~, p=f[i, 1]; if(p%6==1, break)); print1(p", "); Q*=p) \\ Jens Kruse Andersen, Jun 30 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Nick Hobson, Nov 14 2006
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STATUS
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approved
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